International Journal of Education in Mathematics, Science and Technology

Volume 3, Number 3, July 2015, Page 242-261 ISSN: 2147-611X

Using Classroom Scenarios to Reveal Mathematics Teachers’

Understanding of Sociomathematical Norms

Ismail Ozgur Zembat1*

, Seyit Ali Yasa2

1Mevlana (Rumi) University

2Enderun High School

Abstract

The purpose of this study was to uncover the degree to which in-service teachers understand sociomathematical

norms and the nature of that understanding without having to enter and observe their classes. We therefore

developed five classroom scenarios exemplifying classroom interactions shaped by certain sociomathematical

norms. We then administered these scenarios to in-service elementary school and grade 5-12 mathematics

teachers and collected their written responses. We also collected data about what teachers believe about

sociomathematical norms through a Likert-type questionnaire. We then analyzed the data using quantitative and

qualitative techniques. First, the findings suggest that the teachers’ understanding of sociomathematical norms is

neither dependent on the level of schools teachers teach, or their background or demographic characteristics

such as number of years they spent in teaching, specialty area, faculty graduated, highest degree earned, and

gender. Second, what teachers believe about sociomathematical norms seem to be not parallel to how they

analyze the scenarios illustrating sociomathematical norms. Third, use of scenarios was helpful in revealing how

teachers think about sociomathematical norms. Finally, there are three cross-cutting themes to which all the

teachers referred in common for all sociomathematical norms: opposition (opposition to the core of the norm),

social facilitator (considering all targeted norms as supporting and regulating the classroom social environment)

and condition-based (believing that interactions given in scenarios are only possible under certain conditions).

Key words: Sociomathematical norm understanding, In-service mathematics teacher education, Role of

teaching scenarios, Testing mathematics teacher knowledge, Nature of mathematics teacher knowledge.

Introduction

The purpose of this study was to uncover the degree to which in-service teachers understand sociomathematical

norms and the nature of that understanding without having to enter and observe their classes. In so doing we

also aim to understand how the ways in which teachers treat sociomathematical norms differ with respect to

certain demographic variables (e.g., different grade levels they teach, teaching experience, degree owned, etc.).

To answer these questions, we generated classroom scenarios to identify the nature of teachers’ understanding

of sociomathematical norms and the aspects of the classroom interactions teachers focus on in analyzing these

scenarios. Norms can be investigated through long-term classroom observations but the purpose in this study

was to identify teachers’ norm understanding without going into their classes or observing their teaching, which

makes it easier to elicit and interpret their understanding of sociomathematical norms. Following such a method,

we believe, would be useful in conducting effective assessment of mathematics teacher knowledge, designing

useful courses to serve prospective teachers in teacher education programs, and producing practical hiring

policies for administrators.

Theoretical Background, Rationale and Research Questions

Cultural and social processes are intertwined with mathematical activity (Voigt, 1995, cited in Yackel & Cobb,

1996). “The development of individuals' reasoning and sense-making processes cannot be separated from their

participation in the interactive constitution of taken-as-shared mathematical meanings” (Yackel & Cobb, 1996,

p.460). Therefore, learning of mathematics in classes and students’ in-class mathematical activities are integral

* Corresponding Author: İsmail Ö. Zembat, izembat@gmail.com

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to classroom social interaction (Brandt & Tatsis, 2009; Cobb, Jaworski, & Presmeg, 1996). Classroom social

interaction affects both the quality of mathematics learning and student reasoning. Based on this argument and

by benefitting from such a sociological perspective on mathematical activity Cobb and his colleagues developed

the construct of norms, descriptions and examples of which are given below.

Every classroom as a group operates with certain interaction and behavior patterns, which influence quality of

teaching and student learning. When the focus is on the “regularities in patterns of social interaction” (Yackel &

Cobb, 1996, p.460), in general, these patterns of interaction are called social norms; whereas when it is on

“normative aspects of mathematics discussions specific to students' mathematical activity” (Yackel & Cobb

1996, p.461), they are called sociomathematical norms. An example for a social norm is ‘shared solutions needs

to be explained,’ whereas an example for a corresponding sociomathematical norm is ‘solutions should be

articulated through acceptable mathematical explanations.’ In this sense, social norms can be related to any

field, whereas sociomathematical norms are specific to mathematics. Norms, whether social or

sociomathematical, are negotiated among group members (the teacher and the students) and established through

such a negotiation process, not through imposing (Cobb, Stephan, McClain, & Gravemeijer, 2001). As the

group members participate in this negotiation process, they develop taken-as-shared understandings of “when to

do what and how to do it” (Bauersfeld, 1993, p.4, cited in Yackel & Cobb, 1996) in terms of social or

mathematical processes.

Norms, as regularities in behavioral patterns, are apparent in a classroom for an observer regardless of whether

the students or the teacher are aware of them (Voigt, 1996). If a teacher has the necessary knowledge of

classroom norms, then she/he may use it to orchestrate the student–student and student–teacher interactions

(Cobb & Bauersfeld, 1995) as well as to promote students’ mathematical understandings (Yackel, Cobb, &

Wood, 1991). This suggests that knowledge of norms and their effective use should be a core part of teacher

knowledge, which can be considered as part of Shulman’s (1986) description of knowledge of pedagogy. Norms

are also important in terms of helping students develop appropriate mathematical language, support students’

development of mathematical ideas, and design the classroom as a mathematical community that challenges

students’ mathematical reasoning (Yackel & Cobb, 1996). Therefore, teachers’ understanding of classroom

norms may be considered one of the factors affecting the quality of teaching in classrooms.

To further detail the importance of (sociomathematical) norms, why they should be considered as an essential

part of teacher knowledge, and how they impact students’ (mathematical) understanding, we can reflect on the

following scenario. Consider a classroom in which students work on a given problem such as, “what would be

an ‘odd number + odd number’?” and they find their solution using the following solution method: “3 and 5 are

odd numbers and adding them up gives 8, which is even. 7 and 9 are also odds, and their addition gives 16, that

is also even. So two odds, when added up, make an even number.” If the teacher in this class knows about the

sociomathematical norm of justification (giving convincing/acceptable mathematical explanations/rationales),

s/he would follow up on this answer through questions like, “how do you know that two odds always make an

even number? How can we make sure?” and continue through such questioning. This in turn would encourage

students to think about a more convincing argument that goes beyond a trial-error method. If this continues

throughout, for example, a semester, then students will likely begin operating with the understanding that “I

should give answers that convince others.” On the other hand, if such a norm is not in the repertoire of the

teacher of this class, s/he would not be aware of the positive effect of such sociomathematical norm on student

development and, as a result, s/he would just accept the first given answer and move on to another question;

hence, miss the opportunity to create an environment supporting students’ mathematical development.

Therefore, norms are essential parts of teacher knowledge in terms of supporting students’ mathematical

development and shaping the quality of classroom (social) interactions.

How well are teachers prepared to orchestrate their classrooms through effective use of such classroom

(sociomathematical) norms? Do teachers even know about classroom (sociomathematical) norms? One way to

investigate in what ways teachers understand norms and how they use these norms is to conduct long-term

research studies, including extensive classroom observations. In fact, several long-term studies focused on

classroom social or sociomathematical norms and how such norms are established either at school level (e.g.,

McClain & Cobb, 2001; Yackel, 2002) or with teachers (e.g., McNeal & Simon, 2000). Even though these

studies are valuable in understanding dynamics of classroom microculture as well as how student–student and

student–teacher interactions are regulated to promote student understanding, they are mostly long-term studies

and are limited to the classrooms of a few teachers. There are two important issues here for which we need

practical solutions: the necessity of long-term studies and working with a limited number of teachers. To solve

the first issue, the following question first needs to be answered: How can we test teachers’ understanding of

norms without entering into their classes and pursuing long-term studies? Answering this question would help

244 Zembat & Yasa

researchers gain time and wisely speculate about possible teaching trajectories that the teachers may pursue

before observing their actual teaching. The second issue is about developing effective measures of testing

knowledge of norms. If the knowledge of norms is a core part of teacher knowledge, then different ways of

measuring such knowledge and that are applicable to big counts of teachers, not a bunch, in a short time need to

be developed. Doing so would inform administrators in developing appropriate ways to assess teacher

knowledge before hiring the teachers. Assessing teacher understanding in this regard would also inform the

mathematics education field about how such understanding differs in different grade levels. Unfortunately, we

could not find any research in the relevant literature focusing on these issues. This study takes the first stab in

finding solutions to these problems through investigating the following research questions:

1. How do grade 1-12 in-service teachers think about classrooms (described in scenarios) operating with certain sociomathematical norms?

2. What do grade 1-12 in-service teachers believe about use of sociomathematical norms in classes? 3. How does participant teachers’ thinking about sociomathematical norms differ with respect to different

grade levels?

4. What is a useful way to assess teachers’ understanding of sociomathematical norms without having to enter their classes?

The purpose of this study was to answer these questions for teachers working within the Turkish education

system.

Method

This study was designed as a mixed-methods study investigating the ways in which teachers understand

sociomathematical norms using qualitative and quantitative measures. The detailed information about

participants, data collection process and data analysis process are given below.

Participants

Overall, 18–22 in-service public school mathematics teachers from each school level (grades 1–4, elementary

teachers; grades 5–8, middle school mathematics teachers; and grades 9–12, high school mathematics teachers)

totaling 61 participated in the study (see Table 1 for demographics). Participant teachers’ ages range from 27 to

58 (average = 37.6, median = 36). Table 1 suggests that participants have enough classroom teaching experience

in order to be able to analyze and talk about classroom scenarios.

Table 1. Demographic information for participant teachers.

We used convenience sampling for choosing participants. The second author went to certain schools in a

metropolitan city in the middle part of Turkey, informed mathematics teachers about this study, and asked them

to voluntarily participate in an approximately one-hour problem-solving session. Then, the author applied the

questionnaire to the volunteers during their spare time in school. Once each individual teacher completed the

questionnaire, the author collected the participants’ written responses and the data collection ended.

Data Collection Instrument

We initially identified seven sociomathematical norms that cut across the relevant literature (Cobb, Stephan,

McClain, & Gravemeijer, 2001; Kazemi & Stipek, 2001; McClain & Cobb, 2001; McNeal & Simon, 2000;

Tatsis, & Koleza, 2008; Van Zoest, Stockero, & Taylor, 2012; Yackel, Cobb, & Wood, 1991; Yackel & Cobb,

1996) and prepared a classroom scenario for each norm to test teachers’ understanding of these norms. We

piloted these early scenarios on 15 mathematics teachers from different school levels and revised them based on

our experience. We then narrowed down the number of sociomathematical norms to five (hereafter, abbreviated

Teaching Experience (in years)

School Level 3-10 11-15 16-20 21-30 Total

Elementary 4 6 8 3 21

Middle School 11 3 1 3 18

High School 4 11 6 1 22

Total 19 20 15 7 61

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as SMN) to be tested and again revised the classroom scenarios that we had previously generated. These five

norms appear vastly in the literature and these norms we believe are the kind of SMN that a mathematics teacher

easily experience in daily teaching practice. Next, we had another PhD mathematics educator check the

questionnaire consisting of scenarios as an outsider and then finalized it based on that feedback. Each scenario

was placed on a single page in the questionnaire (see Appendix 1 for details about targeted SMN and scenarios).

The pilot study showed us that the questionnaire required about one hour completing, and this was enough time

commitment on the part of the participants given that their participation was voluntary.

The SMN we decided to focus on in this study are;

i. offering/distinguishing mathematically different solutions through an analysis of what makes one solution mathematically different from another,

ii. offering/generating mathematically more sophisticated solutions than the given one, iii. reaching a consensus through necessary mathematical argumentation, iv. giving convincing/acceptable mathematical explanations/rationales, and v. using mistakes as opportunities to reconceptualize a problem.

Note that we developed each scenario to help participants specifically focus on the core idea for the targeted

SMN only. For example, for a mathematical difference norm (SMN#1), we gave participants a scenario that

specifically includes a student discussion about what makes a given solution to an addition of whole numbers

problem mathematically different from another. The scenario for the mathematical justification norm (SMN#4)

was also specific to convincing mathematical argumentation – a group of students trying to convince each other

about which of the two given fractions is bigger as if they explain it to a blind person (see Appendix 1 for

further details about scenarios). Our purpose in developing and using these scenarios was to understand how

participant teachers with a variety of backgrounds analyze classroom situations in which students and the

teacher operate with certain SMN and to understand nature of their understanding as reflected in those analyses.

Through such an analysis we tried to develop useful qualitative and quantitative descriptions of how teachers

think about such classroom organizations and what they think they would actually do in terms of SMN

establishment in their own classrooms. In addition to scenarios we also developed a Likert-type mini

questionnaire including two questions with five subquestions for each to also test what teachers’ actually believe

about norms. The details of this questionnaire are given in Results section.

Data Analysis Process

The data analysis process started with organizing the data by giving each respondent a code organizing his or

her responses (Miles & Huberman, 1994) and analyzing those as in Table 2. We also developed rubrics to

analyze participants’ level of understanding of SMN (see Appendix 2 for details). After the first run of the

analysis is completed, we then checked each participant teacher’s responses again and searched for overarching

themes describing teachers’ work for characterizing their understanding of each individual norm as well as

crosscutting themes for norms in general. We generated an initial list of overarching themes, crosschecked them,

and through use of this cycle in iteration narrowed the list in each run of the data analysis down to its final form.

Table 2. Structure used in the initial data analysis process.

Categories used in the first run of data analysis

What the

participant

said

Analysis of

what the

participant said

and level of the

participant’s

SMN

understanding

What issues

regarding

SMN stand

out in what

the participant

said

Participant’s

response to

Likert-type

question 1

Participant’s

response to

Likert-type

question 2

Does the

participant

consider the

norm from a

social or

mathematical

perspective?

Results

Quantitative Results

In analyzing the data, we initially developed rubrics based on data and then we gave each response a score from

1 to 4 characterizing each teacher’s level of understanding about SMN (see Appendix 2 for rubrics). We then

added these scores and ranked them using SPSS 21 (IBM Corp., 2012). Next, we found the average scores for

246 Zembat & Yasa

each separate group (elementary, middle school, and high school) and calculated the differences between

average scores and ranked scores. We then conducted the non-parametric Levene’s test of these difference

scores to verify the equality of variances in the samples (homogeneity of variance) (Nordstokke & Zumbo,

2010; Nordstokke, Zumbo, Cairns, & Saklofske, 2011). The results (Levene statistic=.621; p=.541>.05) show

that the data set has homogenous variance. We finally conducted an ANOVA on difference scores using the

elementary, middle-, and high-school groups as independent variables and found that there is no significant

difference between the groups (F=1.096; p=.341>.05). This suggests that SMN understanding level does not

seem to significantly differ for elementary, middle-school, or high-school mathematics teachers. We also

checked the data to see whether the norm understandings differ based on gender, number of years in teaching,

faculties graduated, specialty area, and the highest degree earned as detailed in Table 3. In all these checks, we

could not find any statistically significant difference among groups.

Table 3. ANOVA results with respect to certain demographic factors.

ANOVA

F p

Gender 1.096 .341

Number of years in teaching 1.983 .084

Faculty graduated .700 .595

Specialty area .995 .446

Highest degree earned 1.164 .320

All these quantitative findings suggest that norm understanding of grade 1-12 in-service teachers does not seem

to change over years and it is not dependent on how strong a mathematical or educational background they have.

We also analyzed teacher responses based on whether they analyzed the given scenario from a mathematical

point of view (e.g., talking only about mathematical aspects of the given scenarios, solving the mathematical

questions given in the scenarios even though not asked) or a social point of view (e.g., considering interactions

as supporting classroom discussions, participation). Interestingly, many teachers consider the given scenarios

for SMN#2 and SMN#4 from a mathematical standpoint as illustrated in Table 4.

Table 4. Treatment of scenarios from a social/mathematical standpoint.

SMN #1 SMN #2 SMN #3 SMN #4 SMN #5

Math Social Math Social Math Social Math Social Math Social

Elementary (1-4) 6 14 18 3 7 12 18 0 12 9

Middle School (5-8) 7 11 18 0 6 12 18 0 11 4

High School (9-12) 13 9 22 0 6 16 22 0 15 5

Table 5. Participant teachers’ level of norm understanding.

Understanding for Level of

understanding Elementary Middle School High School

SMN #1 Very low 7 (35%) 8 (44%) 11 (50%) Low 12 (60%) 9 (50%) 7 (32%) Medium 1 (5%) 0 4 (18%)

High 0 1 (6%) 0

SMN #2 Very low 18 (86%) 7 (39%) 7 (32%) Low 1 (5%) 7 (39%) 12 (55%) Medium 2 (10%) 3 (17%) 3 (14%)

High 0 1 (5%) 0

SMN #3 Very low 9 (47%) 12 (68%) 12 (55%) Low 4 (21%) 4 (22%) 8 (36%) Medium 6 (32%) 2 (10%) 2 (9%)

High 0 0 0

SMN #4 Very low 3 (17%) 4 (22%) 2 (9%) Low 15 (83%) 11 (61%) 18 (82%) Medium 0 3 (17%) 2 (9%)

High 0 0 0

SMN #5 Very low 12 (57%) 8 (53%) 9 (45%) Low 4 (19%) 6 (40%) 9 (45%) Medium 4 (19%) 0 1 (5%)

High 1 (5%) 1 (7%) 1 (5%)

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Surprisingly, the scenarios generated to test SMN#2 and SMN#4 were almost exclusively interpreted by the

participants from a mathematical standpoint; whereas the scenarios targeting SMN #1, #3, and #5 were analyzed

by focusing mostly on the social aspects. In fact in each scenario the participants were specifically asked about

the “normative aspects of mathematics discussions specific to students' mathematical activity” (Yackel & Cobb

1996, p.461) as opposed to “regularities in patterns of social interaction” (Yackel & Cobb, 1996, p.460). This is

also interesting in the sense that Scenarios #2 and #4 were about evaluating mathematical solutions, whereas the

others consist of classroom interactions. This may suggest that the participant teachers consider classroom

interactions as social events as opposed to environments supporting mathematical development of students.

However, this result needs to be justified through further research.

We also checked the degree to which participant teachers understand SMN (see Appendix 2). As illustrated in

Table 5, many of the participant teachers interpreted the given scenarios by operating with a “very low” or

“low” level of understanding of norms. For each scenario designed to test a single norm (except SMN#3 for

elementary teachers, 68%), interestingly, 76%–100% of teachers fall into either the very low or low category in

terms of their understanding of sociomathematical norms. This is quite a big percentage when knowing about

SMN and using them as effective learning and classroom design tools are important assets of teacher knowledge

(Even & Tirosh, 2002). On the other hand, only 5 out of 61 teachers (1 for SMN#1, 1 for SMN#2, and 3 for

SMN#5, totaling 8%) seem to have a high level of understanding of sociomathematical norms, which is quite

low. The participants were also asked, right after their work with scenarios was completed, about what they

believe about the kind of abilities regarding SMN students should have and about whether they allow certain

SMN in their own classes as illustrated in Table 6. Note that each question in each section of the questionnaire

given in Table 6 corresponds to a SMN presented in each scenario given in the questionnaire; however, the

order in which these questions were given to the participants was not same as the one given in Table 6.

Table 6. Likert-type questions asked the participants to reveal what they believe about SMN

What abilities do you think students should have?

Answer this question based on the scale given on the right and

mark the necessary box given next.

1: Absolutely disagree; 2: Disagree;

3: Not sure; 4: Agree

5: Absolutely agree

1 Be able to predict whether the proposed mathematical

solutions are mathematically different from each other. 1 2 3 4 5

2 Try to / Develop more sophisticated mathematical solutions

than the shared solutions. 1 2 3 4 5

3 Be able to reach a consensus through challenging each other’s

mathematical reasoning and convincing each other. 1 2 3 4 5

4 Be able to provide convincing mathematical justifications as

they explain an issue. 1 2 3 4 5

5

Be able to draw lessons from an analysis of the mathematical

mistakes they fall into and modify their knowledge based on

this analysis. 1 2 3 4 5

How often do you allow the following in your classes?

Answer this question based on the scale given on the right and

mark the necessary box given next.

1: Never; 2: Seldom; 3: Sometimes

4: Often; 5: Always

1

In my classes, I provide my students with the opportunities to

understand what makes a solution mathematically different

from another 1 2 3 4 5

2

In my classes, I build the necessary environment for my

students to generate mathematically more sophisticated

solutions than the shared ones. 1 2 3 4 5

3

I provide the necessary environment for my students to

seriously question the shared mathematical ideas and

reasoning to reach a consensus. 1 2 3 4 5

4 I encourage my students to provide convincing and

explanatory arguments in my classes. 1 2 3 4 5

5

I provide my students with the opportunity to use mistakes as

occasions to reconceptualize problems and reorganize their

knowledge by learning lessons from those mistakes. 1 2 3 4 5

Surprisingly, Table 7 shows that majority of the participants who were at a low or very low level in terms of

their norm understanding chose “agree” and “absolutely agree” to comment that students should have those

248 Zembat & Yasa

abilities, and they design their classes to support SMN given in the questionnaire. This contradiction between

participants’ level of SMN understanding and what they believe about SMN suggests that they believe that

SMN are important and should be established without genuinely knowing the core of what it means for a class

to operate with certain established SMN. This contradiction also suggests that teachers’ beliefs do not guide

their classroom practice; they believe but do not operate with that belief. Though, we are cautious about this

result since it should be tested through further research.

Table 7. Teachers’ norm understanding versus their beliefs about norms.

Understanding

for

Level of

understanding

What abilities do you think students

should have? (see Table 6)

How often do you allow the following in

your classes? (see Table 6)

Agree Definitely agree Total* Often Always Total**

Norm #1 Very low 18 (30%) 6 (10%)

83% 15 (25%) 5 (8%)

66% Low 15 (25%) 11 (18%) 14 (23%) 6 (10%)

Norm #2 Very low 24 (39%) 6 (10%)

80% 19 (31%) 5 (8%)

63% Low 10 (16%) 9 (15%) 10 (16%) 5 (8%)

Norm #3 Very low 20 (33%) 11 (18%)

76% 19 (31%) 5 (8%)

59% Low 12 (20%) 3 (5%) 7 (12%) 5 (8%)

Norm #4 Very low 6 (10%) 2 (3%)

79% 6 (10%) 2 (3%)

66% Low 25 (41%) 15 (25%) 23 (38%) 9 (15%)

Norm #5 Very low 20 (33%) 6 (10%) 75% 15 (25%) 8 (13%) 56% Low 12 (20%) 7 (12%) 13 (21%) 2 (3%)

* This total represents the total percentage of participants who “agree” or “definitely agree” with the given statement.

** This total represents the total percentage of participants who chose “often” or “always” as their answers.

To sum up, quantitative analysis helped us see, in addition to the contrast between the teachers’ level of

understanding of SMN and what they believe about SMN, that SMN understanding of participant teachers does

not change with respect to the grade level teachers teach and other aforesaid demographic variables. Based on

these findings we looked through all data to examine more deeply to uncover the nature of teachers’

understanding of SMN, which is detailed in the following section.

Qualitative Results

We conducted a qualitative analysis to better understand about the nature of teachers’ understanding of SMN to

characterize their level of understanding of each targeted SMN. Our analysis suggests that the teachers view

each targeted SMN as in Table 8. We initially identified what aspects of the given scenarios the teachers

highlighted in analyzing those scenarios through which we characterized their level of understanding of SMN.

In the following section, we initially categorize teachers’ descriptions of SMN as in Table 8 and we then defined

and detailed each category of descriptors by drawing on the relevant data.

Table 8. Teachers’ characterization of sociomathematical norms. SMN #1 –

Mathematical

difference norm

SMN #2 – Mathematical

sophistication norm

(5% did not comment)

SMN #3 –Mathematical

consensus norm

(4% did not comment)

SMN #4 –

Mathematical

justification norm

(14% did not

comment)

SMN #5 – Mistakes-

as-opportunities norm

(5% did not comment)

Unusual (%33) Easy/Understandable

(52%)

Social negotiation tool

(30%)

Understandable

(23%)

Correction (23%)

Social supplier

(%28)

Making connections

(16%)

Develops student

thinking (38%)

Numeric pattern

(26%)

Support social

environment (26%)

Restrictive (%23) Dependent on

student/class level

(11%)

Conditional (23%) Structural (16%) Mathematical content

(23%)

Variety (%16) Supporting learning

(16%)

Consensus (5%) Non-explanatory,

non-convincing

(21%)

Teacher centered

(23%)

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1. Teachers’ Use of Descriptors for Mathematical Difference Norm

In the mathematical difference norm (SMN#1) scenario students decide on whether to accept or reject the

offered solutions to be written on the board based on their analysis of mathematical similarity or difference of

these solutions. Participant teachers responded to this scenario in four different ways.

a. Some of the participants (33%) considered this way of interaction of students and what makes a solution

mathematically different from another by resorting to the following descriptions: “effective solutions,” “beyond

routine solutions,” “solutions paving the way for making mental calculations,” “memorable solutions,” “nice

solutions,” “interesting solutions,” “solutions helping develop different perspectives,” “wise solutions,”

“solutions having structural difference,” “solutions contributing to [students].” All these descriptors suggest that

these teachers seem to understand that the given scenario is about students’ elimination of proposed

mathematical solutions. These teachers considered this way of interaction as singling out unusual solutions and

disregarding others. The reason we called this understanding as unusual is that teachers in their responses

constantly referred to solutions beyond expected, different or promoting different abilities.

b. Another group of participant teachers (17, 28%) considered the way students in the scenario approach the

given solutions as both improving students’ individual and social identities and establishing and developing

classroom social setup. These teachers focused on the idea that such interaction supports students’ social

development as members of the class. For example, teacher responses in this category included that such

interaction improves certain social qualities of students (“self-confidence,” “awareness,” “success,”

“practicality”) and frees student thinking from mechanic procedures (“being away from memorizing facts,”

“brain storming,” “encourage to think”). Since these are the qualities that are not directly about mathematics but

support students’ social development, we call this way of characterization the teachers made as social supplier.

In addition, 13 of these teachers focused only on how such interaction affects the social setup of the classroom.

What makes this interaction valuable for these teachers were the fact that such interaction “provides

opportunities for students to learn from each other through discussion,” “enables consensus among students,”

“develops and organizes social environment,” “provides democratic environment,” “helps students respect each

other,” “increases classroom participation,” and “helps reach a conclusion together.” All these descriptions are

about social aspects of classroom or student development. Teachers in this category put forward these aspects as

opposed to the potential mathematical contribution of the interaction to the students in the given scenario. In

other words, they considered a sociomathematical norm as a social supplier for students rather than a

mathematical one.

c. The third group of the participants (14, %23) equates focusing on mathematical difference in the given

scenario to focusing on solutions that are “appropriate for students’/class’ level,” “doable,” “easy,” “useful,”

“reasonable,” “not explained well,” and “challenging.” In this sense, teachers in this category considered

mathematical difference as restrictive to certain qualities. For example a teacher suggested,

“I agree. I would behave the same way as the teacher did. I would listen to any kind of answer, I would

choose the one that provides the maximum gain with respect to the understanding to be learned

(rounding to the nearest ten), and I would have them written on the board […].”

This perspective of the teacher suggests that his focus was on choosing the solution providing the “maximum

gain” instead of mathematically different solutions. In so doing, the teacher restricts mathematical difference to

a limited domain, the solutions providing the maximum gain. Some of the teachers in this third group (3, %5)

completely opposed to focus on different mathematical solutions since they might cause “confusion” among

students and might have “discouraging” effect on students whose solutions are disregarded because of

similarity. Moreover, 6 of the teachers in this group (10%) also believe that the teacher should be the determiner

of whether given solutions are same or different, not the students. All these descriptions of participants suggest

that teachers in this category either considered focusing on mathematical difference as restrictive for the class or

suggested focusing on a limited number of solutions without looking into mathematical difference.

d. The fourth group of participants (10, 16%) approached this scenario by pointing out that all given solution

methods should be shared with everyone; “I would tell students that I support all the answers.” They supported

this idea with the rationale that this would give students the opportunity to see the variety of solutions all at once

regardless of whether they are similar or different. For these teachers mathematical difference means seeing

variety of solutions.

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2. Teachers’ Use of Descriptors for Mathematical Sophistication Norm

In the given scenario for mathematical sophistication norm (SMN#2) four different solutions are given to

participant teachers and they were asked to decide on with which one they would engage their students in their

classes. The way the participants interpreted the given scenario is detailed below.

a. About half of the teachers (32, 52%) who answered this question pointed out that the solution to be focused

on and used for encouragement in classes should be “clear,” “understandable,” “easy,” “concrete,” “practical,”

“basic,” “less likely to make mistakes,” and “easier to comprehend” instead of being sophisticated furthering

student thinking. For example, one teacher suggested “I would prefer the answer of group #1. It is easy to

understand and easy to explain. Having fewer calculations make its expression easier”, whereas another teacher

wrote, “I would prefer the answer of group 3, because I think that it is a figure that is understandable by more

students easily.” Such descriptions suggest that teachers make reference to easiness/understandability for the

notion of mathematical sophistication.

b. Some of the participants (10, 16%) preferred solutions that have the potential to enable students to make

connections among upcoming concepts to be pursued in the curriculum or that have the potential to be

transferred to other upcoming concepts or grades. These teachers particularly pointed out the solutions they

would focus on as the ones “having students comprehend different mathematical relationships, applicable to

other topics,” “giving opportunity to prepare [students] for upcoming concepts [to be taught],” “including

mathematical relations,” and “transferable to other problems.”

For example, half of the teachers in this group chose second group’s answer in the given scenario since it

includes difference of squares as illustrated in the following typical answer:

Teacher: I would prefer brainstorming and thinking more on Group 2’s solution because the topic

under question is about squares. It is important since it is about finding square of a number and about

understanding the relationship between squares of numbers. At the same time it is also important with

respect to establishing a relation among visual patterns.

This teacher chose second group’s answer because it directs student attention to relations among visual patterns

and in-between squares of numbers. Hence, the teachers in this group make their preferences based on making

connections.

c. Another group of teachers (7, 11%) made their decision about the given solutions based on class’ or students’

level of understanding. The participants in this category in some way or other mentioned that “each solution

method can be applied in different level of classrooms.” These teachers think that the solution to be pursued in

classes is dependent on student/class level.

d. The last group of participants (10, 16%) in this group considered the solutions to be brought to students’

attention in the class as the ones that give students the opportunity to learn better, in other words, that support

student learning. Seven of these teachers used descriptions for the solutions as “useful,” “visual,” “easily

stored in mind,” “deepens student thinking,” “interesting,” “long lasting,” and “giving opportunities to

reorganize knowledge,” for solutions to be stimulated in class.

In this regard, teachers in this category consider sophisticated solutions as solutions consisting of elements

supporting student learning. One of the teachers, who gave the only level-4 answer in this category, pointed out

the reason for his preference of second solution as follows:

Teacher: I would encourage students to think about and brainstorm for second group’s answer. I

would focus on the second group’s answer among others in order to direct my students to produce

higher level and advanced mathematical solutions. Once they conceptualize such a solution, upon

running into a question like that, it would be easier for my students to solve it using the same method.

Even though the quality of this answer is much higher than the other 9 teachers, based on what he said we can

conclude that this teacher’s purpose is about supporting student learning, like others.

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3. Teachers’ Use of Descriptors for Mathematical Consensus Norm

The scenario generated to test teachers’ understanding of mathematical consensus norm (SMN#3) included

students’ reaching a consensus through debating their ideas. Teachers’ use of descriptions for this norm fall into

four categories as detailed below.

a. A number of participant teachers (18, 30%) considered the interaction pattern in the given scenario as a social

negotiation tool. For these teachers students in the given scenario reach “the right” or “most appropriate”

solution through “interaction,” “discussion,” “questioning,” “inspecting,” and “exchanging ideas.” Some of

these teachers here (14, 23%) also referred to the facts that students in the given scenario “interpret the reasons

of given ideas,” “reveal reasoning behind ideas,” “defend” and “prove” given ideas and “convince each other”

and these teachers saw the classroom operation valuable with respect to these aspects. Such descriptors as

“discussion,” “inspecting,” “questioning,” etc. are the kind of descriptors modelling the classroom interaction in

the given scenario from a very general perspective without any solid examples from the given scenario. For

example, a teacher said, “students reach the right solution through discussing and inspecting ideas” without

mentioning which (mathematical) ideas and how. In this sense participant teachers found the classroom

interaction in the given scenario useful as a social negotiation tool instead of students reaching a consensus

through a mathematical negotiation process which may contribute to students’ mathematical development.

b. A significant number of teachers (23, 38%) described such classroom operation as “nice,” “preventing

students from rote learning,” “having students learn better,” “enabling new ideas to be constructed on previous

ideas,” “enabling long-lasting learning,” and “making students active.” However, the teachers making these

descriptions about the operating of the class made them in very general terms without providing any examples

or details from the given scenario. As a result participant teachers in this category focused on the contribution of

such class operation to the development of student thinking and valued it using this aspect.

c. Some of the participants mentioned that in order for such a class to operate, there should be certain a priori

conditions established in that class. For example, these teachers mentioned that in order to have such a class

students should have “the necessary prior/background knowledge,” “communications skills,” “culture of

debate,” and “developed reasoning skills.” These participants also added that these conditions must be satisfied

before such classroom operation could take place. Hence these participants seem to consider these conditions as

obstacles to be overcome and it is impossible to have such a class without these conditions for these teachers. In

other words, the teachers in this category did not seem to find such classroom scenario realistic and therefore

tied such operation to certain a priori conditions. For example, one of the teachers mentioned, “Such a

discussion environment is impossible in the classes I teach. I neither can teach this topic this well nor are the

students well-behaved. I do not have students at this level.” As seen in this response the teacher based his

argument on certain conditions such as not having “well-behaved” students or not having students at a certain

“level.”

d. A few participants (3, 5%) pointed out that with the help of the classroom operation in the given scenario

“students are convinced through adding to each other’s ideas,” “students agreed upon a common solution,” and

“students themselves reach a consensus together.” These teachers are the only ones who can identify, even

though in general terms, that there is a negotiation process to reach a consensus among students in the given

scenario. In this sense these three participants referred to reaching a consensus in analyzing the given scenario.

4. Teachers’ Use of Descriptors for Mathematical Justification Norm

In the given scenario for mathematical justification norm (SMN#4) participant teachers are given four different

student solutions to a pattern problem and asked to decide which one was more exploratory and convincing. The

way the participants interpreted the given scenario is detailed below.

a. Some of the teachers in this category (14, 23%) focused on different aspects of the solutions such as

“easiness”, “shortness,” “clarity,” “understandability,” “explanatory,” “comprehensibleness,” “clarity,”

“plainness,” “none confusing,” or “concreteness.” All these descriptors suggest that for these teachers the notion

of “convincing solution” is equivalent to easily understandable solution. These participants also considered

providing convincing arguments as time consuming and at some point in their description pointed out the

importance of solutions being easily understandable.

252 Zembat & Yasa

b. A group of participants (16, 26%) put more emphasis on numerical solutions and pointed out that the more

numerical the solution was the more convincing it would be. Nine of these teachers considered the given

numerical pattern in the solution as “far from rote,” “not confusing,” “not crowded with formulas,” “short and

right,” whereas 7 of them considered the pattern as appropriate for grade 1-8 students only. These teachers

decided on whether a given solution includes appropriate justification or not based on whether it includes a

numeric pattern or not. For these participants numeric pattern inclusion, rather than having the necessary

mathematical argumentation, is what makes a solution justifiable.

c. In the given scenario the first group’s solution was inductive and the second group’s answer was based on

deductive reasoning. Whereas the first group of students gives the formula as a solution to the given problem

and then explains the components of the formula, second group figures out the formula inductively building the

numeric pattern and then finding the formula. 10 teachers (16%) preferred second solution and considered it as

“easily remembered,” “paving the way for reasoning,” “logical,” “easily drawn,” and “working through figure-

formula connection.” One typical answer in this category is, “The second group’s answer is more explanatory

and convincing. Because the solution starts from the beginning, generalize the question and tries to find a

formula. It easily uses the figure and finds the formula inductively.” All these suggest that participants in this

group referred to the structural nature of the given solution and their preference was on inductive nature of the

solution.

d. A group of teachers (13, 21%) equated convincing arguments with that of “not crowded,” “practical,” “most

plain solution,” “not including any explanation,” and “directly giving a clear formula.” The interesting issue

here is that teachers in this group almost equate “convincing” to non-exploratory or non-convincing. In other

words, the better is the solution method, the shorter it gets the student to the formula or the lesser explanation it

includes.

5. Teachers’ Use of Descriptors for Mistakes-as-Opportunities Norm

In the given scenario for mistakes-as-opportunities norm (SMN#5) participant teachers are given a classroom

scenario in which students derive lessons from a faulty solution about a sharing situation and fraction addition.

The different ways in which the participants interpreted the given scenario are detailed below.

a. A group of teachers (14, 23%) used some general descriptions to explain the interaction in the class such as “a

mistake is seen/found/derived by students,” “a mistake is corrected,” “it is written on board so it does not occur

again.” The teachers here considered such classroom work as a way of simple correction (by the teacher) of

student mistakes instead of a regular interaction pattern to catch a problem and reconceptualize it.

b. Another group of teachers (16, 26%) consider such classroom interaction as a way to support social

environment in class. Such interaction for these teachers enable students to “brainstorm,” “discuss,” “question

and investigate,” “be active,” and “have the opportunity to think” to reach the right solution. They also support

such classroom interaction since such environments “teach students tolerate other opinions” and “enable having

different perspectives and finding different solutions.” All these descriptions suggest that the participants in this

category valued this way of interaction because of the opportunities provided to support social environment.

c. The teachers who mostly focused on the mathematical content in the given scenario (14, 23%) interpreted the

teacher’s purpose in the scenario as to identify, or to have students identify, the mathematical errors in the

shared solutions in class and these participants basically described the errors in the given solutions. For

example, the teachers in this category pointed out that the purpose in the class is “to show students that two

fractions cannot be added without finding common denominator,” “to show how to add fractions,” and “to teach

the importance of size of quantities in fraction addition and subtraction.” In fact, in the given scenario, the issue

is about the sharing of a quantity among so many groups as opposed to the addition of fractions. The teachers in

this category only focused on the mathematics involved for the purpose of correcting a mistake but ignored the

given classroom interaction and how that supported student understanding. Hence, they only valued the

mathematical content in the given scenario.

d. The scenario was given to the teachers with two follow-up questions as (a) and (b). A group of teachers’

responses (14, 23%) to these two questions were completely contradicting. For example, a teacher mentioned,

“The teacher aims for his students to find the right solution through discussion of wrong ideas” for part (a),

whereas he said, “I would give each student a cake. I would then partition the remaining cake into 8 pieces and

distribute them” for part (b). First part of this response is about reaching a solution through “discussion”

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whereas the second part is about teacher-centeredness, how the teacher herself would do it through show-tell

method, which is a contradiction. Like this one, the teachers in this category used words remotely connected to

classroom norms in answering the first part of the question, whereas they were in a denial position by resorting

to teacher-directed and teacher-centered classroom teaching mode.

Discussion of Results and Conclusions

The purpose of this study was to understand grade 1-12 in-service teachers’ level of understanding of SMN and

the nature of that understanding as it was reflected in teachers’ analyses of given classroom scenarios in which

the students operate with certain established SMN. The relevant literature includes many studies about norms

through long-term classroom observations, whereas this study targeted revealing teachers’ norm understanding

without observing the teaching of teachers.

The findings suggest that norm understanding is neither dependent on which school level teachers teach nor

dependent on their background or demographic characteristics such as number of years they spent in teaching,

specialty area, faculty graduated, highest degree earned, and gender. This is interesting in the sense that the

more experience teachers have does not necessarily mean the better they would do in classes. This is also

consistent with the results in the relevant literature (e.g., Monk, 1994, Rice, 2010). The fact that norm

understanding does not differ significantly with respect to grade level also suggests that teacher preparation

programs for different levels in Turkey seem to be closely aligned and do not prepare teachers well about norms,

which is an indispensable part of classroom teaching (Voigt, 1994; Even & Tirosh, 2002).

It is also interesting that what participant teachers believe about SMN are almost in contrast with what they

actually know about these norms. They believe that norms are integral to teaching and students should operate

with certain SMN whereas they lack the necessary understanding of SMN. This suggests that teachers need

support in understanding effective ways of orchestrating classroom interaction that support students’

mathematical development, which is an essential part of teacher knowledge as implied by Voigt (1996). In

addition, use of scenarios was helpful in uncovering teachers’ level of understanding of sociomathematical

norms more accurately than Likert-type questionnaires only testing teacher beliefs. There are attempts in the

literature to develop measures, such as multiple-choice questionnaires, to check the quality of mathematics

teacher knowledge (Hill, Schilling, & Ball, 2004). As Ball mentioned “For the field to grow to contribute to

policy and practice, and to teachers’ learning, however, we need to build capacity for smart, probing,

comparative and large scale studies” (Adler, Ball, Krainer, Lin, & Novotna, 2005, p.378) and we intended in the

current study to take a stab at this issue through use of scenarios. The scenarios gave the participant teachers an

opportunity to analyze the nature of the interaction embedded in the scenarios, which in turn helped us identify

teachers’ understanding of normative aspects of mathematical discussions. The follow-up Likert-type

questionnaire, even though it targeted the same norms embedded in the scenarios, gave us the opportunity to see

the gap in between what teachers actually know and believe. Therefore, use of scenarios in revealing teachers’

understanding of sociomathematical norms may be a useful venue to follow in this regard, although finding

more effective ways to analyze teacher responses is necessary.

As we run through the data about teachers’ analyses of the given scenarios in which the classes operate with

certain SMN, we identified three crosscutting themes that the teachers referred to in common for all SMN which

reveal their overall understanding of SMN:

1. There is an opposition to the core of the norm being analyzed. Note that the participants focus on unusual

instead of difference for mathematical difference norm, on easy solutions instead of mathematically

sophisticated questions, on the idea of giving students no explanation for solutions instead of mathematical

justification, on direct teaching of mistakes instead of use of mistakes as opportunities to help students’

development. In so doing, the teachers in a sense focused on the opposite characteristics of norms being

exemplified in the given scenarios.

2. There is this perception among the teachers that SMN are important as more of a social organizer (regulate

student behavior to help them how to act, talk or discuss) for students than as a mathematical developer. They

considered, in one way or the other, all of the targeted norms as supporting the social environment in the classes.

Moreover, teachers mostly analyzed social processes involved in the given scenarios that actually include

classroom interactions, whereas they focused only on the mathematical aspects of the given scenarios in the

absence of those interactions. This may suggest that participant teachers see classroom interactions as social

phenomenon to be analyzed that does not necessarily support students’ mathematical development. Therefore,

254 Zembat & Yasa

the data led us believe that sociomathematical norms for the participants are important only because they

facilitate social aspects of behavioral patterns – as social facilitator.

3. The teachers in this study seem to consider SMN as condition-based. They believe that the kind of interaction

within a class presented in the given scenarios is only possible in classes under certain conditions (e.g., focusing

on restriction of mathematical difference norm, having students with appropriate level of mathematics for other

norms) and with students having certain characteristics (e.g., having respect for others, etc.) and knowledge.

The teachers participated in this study had a range of experience in teaching but seem to lack the necessary

knowledge about SMN to support student development well. This is reasonable and expected in the sense that

these teachers might not have had the opportunity to experience, as a student, the classes in which SMN are

established and therefore could not draw on such experience once they become a teacher. It also seems that

teaching experience was not supportive for these teachers to become aware of SMN. As a result, these teachers

misinterpreted these SMN by resorting to certain common aspects like opposition, social facilitator and

condition-based. The fact that these teachers operate with certain understanding of SMN in analyzing given

scenarios implies that the teacher education programs they went through did not prepare them well enough to

help them “understand what a sociomathematical norm is and construct pedagogical strategies that can be

applied in a variety of contexts” (Kazemi & Stipek, 2001, p.135). Such lack of understanding seems to lead the

teachers to separate “individuals' reasoning and sense-making processes … from their participation in the

interactive constitution of taken-as-shared mathematical meanings” (Yackel & Cobb, 1996, p.460). This in turn

would affect student understanding and the variety of opportunities that the teachers may introduce in classes.

Therefore, in order to adopt a sociological perspective as an integral part of mathematical activity and benefit

from such perspective to support students’ mathematical development, SMN should be seriously considered as a

core part of mathematics teacher knowledge. In this regard, designing or developing appropriate courses for

teachers and teacher education programs in a way that help teachers develop solid understandings of SMN is

important and should be seriously taken into account by mathematics educators and policymakers.

Acknowledgements

The data used in this study is from the master thesis work of the second author (Yaşa, 2015) that was supervised

by the first author from Mevlana (Rumi) University. Part of the data used here is presented in the following

conferences during 2015: 24th

National Educational Sciences Congress (Niğde/Turkey), International

Conference on Education in Mathematics, Science & Technology (Antalya/Turkey), and Turkish Computer and

Mathematics Education Symposium (Adıyaman/Turkey). In addition, we would like to thank the anonymous

reviewers for their thoughtful and timely reviews that helped improve the paper significantly.

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Appendix 1 – Copy of the Norm Questionnaire

Section I

This section was designed to ask about demographic information for teachers.

Section II

Norm #1: Offering/Distinguishing mathematically different solutions through an analysis of what makes one

solution mathematically different from another.

Scenario #1

Miss Aisha asked her students to do “18+32” mentally and told them that she would put up the responses on

which the class agreed on the board. As you will see in the following dialog, the students valued and accepted

some of the answers, whereas rejected others and the teacher allowed for such an interaction.

a. What kind of a perception/understanding should there be as to how mathematical solutions should be

ingrained in the students in this class so that they deemed some shared answers valuable to be written on the

board whereas they rejected the others?

b. If you were the teacher in this scenario, explain in detail whether or not you would support the way students

themselves evaluate the shared solutions in this way.

[Note: As you answer the question think about the given scenario here as opposed to the class you currently

teach. Instead of giving answers such as “I’d support it” or “I’d not support it,” give a justification for your

answer.]

Teacher: I would like you to share your solutions with your classmates. Once you discuss your

answers, I will put the ones you valued up on the board. Aslı?

Aslı: I added up 18 and 32 using the known column method. In this way the answer is 50!

Teacher: Alright, you heard Aslı’s answer. Is that right? [students nod their heads to approve the

answer] Is there anyone who did it in another way?

Osman: Well, I also did write 32 first and put 18 right under that and added them up from top to

bottom.

Fahri: [by talking without permission] I did add 32 and 18 side by side. My answer is also 50.

Ayça: [Talking to Osman and Fahri] Good but how do your answers differ from Aslı’s? You both

add mentally either side by side or through column approach.

Ömer: [with a tone of supporting Ayça] I think so too! All these three answers are alike. Addition

rule is used in all of them. If we were to put them up on the board we should only write one

of them because they are all done using same logic.

Teacher: Do you all agree with this? What do you think?

Class: [talking together] Yeees!

Teacher: Alright! I will put Aslı’s method up on the board. Is there anyone who did it in another

way? [choosing one who raised her hand] Betül?

Betül: Instead of adding 18 and 32 I added 20 and 32 since it is easier, which makes 52. Then I

subtracted 2 from this result because I’d added 2 to 18 initially. The result is 50.

Osman: This solution is nice. We can write it on the board.

Cemil: [interrupting impatiently] Betül’s seems to me different too. I haven’t thought of that. I first

added tens, 10 plus 30 makes 40. Then 8 and 2 makes 10. 40 plus 10 makes 50. I found it

this way.

Teacher: You heard about Cemil’s and Betül’s solutions. Which one do you think we should put on

the board?

Aslı: There are no such solutions on the board. They are different! I think we should put both.

Fahri: I think so too. One is adding up tens whereas the other adds and subtracts 2. [Other students

in the class also agree on this idea and then the teacher put both answers on the board.].

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Norm #2: Offering/generating mathematically more sophisticated solutions than the given one.

Scenario #2*

A teacher showed a picture similar to the one on the right to the students

and asked, “Find a way to figure out how many unit squares there are in

the shaded area in this picture without counting one by one” and then he

found 4 types of answers in students’ notebooks. Without making any

comment, the teacher put the answers on the board as given next. If you

were the teacher in this class, once the following answers were given by

the students, especially which one of those answers would you have

your students think more and hard on? Explain your reasons in detail.

[Note: Here you are not asked to explain what kind of a teaching you

would do or which solution you would explain to the students. Instead,

you are asked to explain which one of these 4 answers, given by students

themselves, you would have your students focus on.]

Type-I Answers

There are 8 units on each side except the corners. In this

case, there are 4×8=32 square units on each side except

the corners. Since there are 4 square units on the corners

[in total], there are 32+4=36 square units altogether.

Type-II Answer

We have 102=100 square units in total. There are

82=64 unit squares inside except the shaded

border. 100-64 = 36 would give us the number of

units in the shaded border just like in a2-b

2.

Type-III Answer

There are 2×10=20 units on the left and right vertical

sides (including the corners). There are 2×8=16 units at

the top and bottom. In total, there are 20+16=36 units in

the shaded border.

Type-IV Answer

When we count the differently colored units on

the border by thinking successively of 9 units at a

time, we see that there are 9+9+9+9=36 units in

total.

*Note that the question of finding the number of shaded square units laid in the border of a 10 by 10 grid was

first seen in the video case given in Boaler and Humphreys (2005) and adopted from this source.

258 Zembat & Yasa

Norm #3: Reaching a consensus through necessary mathematical argumentation.

Scenario #3

A teacher asked, “which of 4/5 and 6/7 is bigger?” to her class where students do not know equivalent fractions

and decimals. She also wanted the students to explain their answers as if they are to explain them to a blind

person. She separated the class into three groups and the following discussion took place among these groups.

a. In what kind of a classroom in which there is such an established understanding/perception of in-class

argumentation would you see such an interaction among students? In answering this question, analyze the dialog

given next based on the kind of reasoning, questioning, inquiry, and answers students used.

b. Why do you think the teacher in this scenario allows such an interaction among students? Would you follow

such a method in your own classes? Explain your answer.

[Note: Consider the question as if there is no competency (in the negative way) among students in this class.]

Group I: We think that “6/7” is bigger. Because when we draw both fractions, “6/7” looks bigger

than 4/5. Since it looks bigger, “6/7” should be bigger!

Group II: Yeah but a blind person cannot see your drawing. So this cannot be a complete solution.

We think that both fractions are same, because the difference between numerator and

denominator is same for both fractions, namely, 1. Since differences are same, the

fractions must be same. What do you think?

Group III: [Talking to Group II.] You did not accept Group I’s solution but we wonder how accurate

your solution is. Numerator–denominator differences for both “99/100” and “1/2” are 1

too, but are these same. Try to draw those and you will see what we mean. So we should

not put the drawing aside.

Group I: They are right, their answer is nice! For example “1/2” is a half and “99/100” is a fraction

that is very close to 1, so it is much bigger than “1/2”; in fact, there is no need to draw it.

Well, we wonder if it doesn’t matter at all if the difference between numerator and

denominator is 1.

Group III: The fact that the difference is 1 may be important. If we were to draw, what would we do

– let’s think about it. We may then be able to visualize it in our minds. If we were to draw

these fractions, we would shade in 4 parts out of 5 from a whole; whereas we would

shade in 6 parts out of 7 from the same whole.

Group II: That’s it, we found it! There is only one piece left unshaded for each fraction. In other

words, the difference between numerators and denominators is “1” for both fractions.

Instead of focusing on the shaded parts what if we focused on the unshaded parts? In

both, there is a single part unshaded.

Group I: In that case, both fractions are one part away from the whole. We can make this

conclusion from the unshaded parts. Which fraction would be further away from the

whole?

Group III: Don’t we need to think about the size of the parts? In which fraction are the parts bigger?

Group I: Fifths are bigger than sevenths.

Group II: How do you know? Don’t tell us that you drew again!

Group I: We can also draw but there is no need for that, it is easy for us. We divide a whole into

five parts when we find fifths, but we divide the whole into 7 parts when we find

sevenths. So when we divide the whole into seven, the parts will be smaller.

Group II: Since 1/7-part is smaller than a 1/5-part, then isn’t 6/7 greater? Because 6/7 would be

closer to the whole. Since 1/7 is a smaller part, then 6/7 is closer to the whole.

Group III: We think you are right! Since the 1/5-parts within 4/5 are bigger parts, then the fraction

4/5 is further away from the whole.

Group I: We solved it, we guess! [The other groups agree either through nods or through “yes”]

259

IJEMST (International Journal of Education in Mathematics, Science and Technology)

Norm #4: Giving convincing/acceptable mathematical explanations/rationales

Scenario #4**

Figure 1 Figure 2 Figure 3

As seen earlier, some shapes were created using the unit cubes, respectively. Assuming that the

same method continues, how many cubes does Figure 17 consist of? How many cubes does

Figure n consist of?

The earlier question was asked in a class, and four different student groups gave four kinds of answers as

shown next. If you were the teacher in this class, which one of the following answer(s) given by students

would you find explanatory and convincing with respect to content, given explanations, and narration and

encourage your class to give such answer(s)? Why?

[Note: Consider for this question that the students in the scenario have the necessary knowledge level to

understand and talk about given answers and explanations. You are not explaining these answers to your

students; on the contrary, the answers are already given by the students. Here you are only asked to explain. As

a teacher in this scenario, what kind of answer(s) would you encourage in your classes and why?]

Answer of Group 1

The number of cubes in each step is 5(n-1)+1.

Here, n gives the order of Figure (1 for Figure 1,

2 for Figure 2, etc.). For example, in Figure 3,

there are 5 arms, each of which has 2***

(or ‘n-1’)

cubes, so it makes 5×(n-1). In each figure, there is

one extra cube in the center, so this gives us the

result of [5(n-1)]+1. As a result in Figure 17,

there are 81 cubes.

Answer of Group 2

There is a single cube in Figure 1, 2 cubes rising in the

center of Figure 2, 3 cubes in the center of Figure 3, 4

cubes in the center of Figure 4, etc. – so in Figure n, there

are n cubes in the center. Besides, there are 4 arms coming

out of the center in all figures except the first one and there

is one less number of cubes than figure order in each arm –

or the number of cubes in each arm is 4×(n-1). So, the

answer is n+4(n-1). There are 81 cubes in Figure 17.

Answer of Group 3

There is one cube in Figure 1, and 5 cubes are

added to this one in making Figure 2. When

figures are created, the number of cubes is

increased by 5 each time. If we continue like this,

we would find 81 cubes in Figure 17.

Answer of Group 4

In the given figures (except Figure 1), each of them

increases with five arms but in each figure there are 4

cubes missing. Therefore, 5n-4 gives the number of cubes.

As a result, there are 81 cubes in Figure 17.

**Note that the question of finding the number of cubes in a sequence is adopted from Van Zoest, Stockero and

Taylor (2012).

*** This number was mistakenly given as “4” in the actual questionnaire; however, participants did not catch

(or purposefully ignore) this minor mistake and evaluated the solutions correctly. This may also suggest that

teachers did not pay attention to the details in the given solutions but to the consistency between formula given

in each solution and the result for Figure 17.

260 Zembat & Yasa

Norm #5: Use mistakes as opportunities to reconceptualize a problem

Scenario #5****

A teacher asked her students to equally share 9

cakes among 8 people and students gave their

answers after they worked on the question for a

while. First, the teacher herself silently observed

one group’s solution, as shown on the right.

As the students in this group solved the problem, they thought about the pieces numbered 1, 2, …, 7, 8 as the

halves to be given to each person. Once they shared 4 cakes among 8 people as halves, they separated each of

the remaining 5 cakes into 8 equal parts and took 1/8 part per person [check the shaded triangular pieces in the

drawing]. In this case, each individual got half a cake in addition to 1/8-part they had taken from each of the

remaining 5 cakes. So each individual gets “1/2+5/8” cakes. Based on this work, this group found their answer

as 6/8. In so doing, they actually thought that 5 of 1/8-parts makes 5/8, which is correct, but they also thought

about the half that each person gets as 1/8 and mistakenly thought that each person gets 6/8 cakes in total.

The teacher put this answer, explained silently to the group, up on the board without making any comment.

Then, the following dialog took place in the class.

Teacher: One of the groups drew this diagram and found the answer as 6/8. Do you agree with

this answer? First think about this. Is there any reason for you to not agree? If there is,

discuss this issue at your tables and arrive at a conclusion.

Cemile: I don’t believe that it is 6/8, it cannot be 6/8. Because there are 5/8s here [pointing to 5

shaded triangles], do you see that? And there is a half here [pointing to the halves coded

as 1,2,3,…8]. These quantities in total need to exceed 6/8!

Kamil: I think 6/8 is correct. Everyone got 6 parts – for example, the first person got the part

numbered as 1 and 5 shaded triangles, namely he got 6 parts in total.

Özgür: Are these parts same? They don’t look same.

Cemile: Yes, 6/8 is not correct! Because look, there is 5/8 here [referring to 5 shaded triangles].

There is 1/2 here [pointing to the half numbered as 1]. That’s why the answer is 9/8. So

the answer to the question is 9/8 instead of 6/8. Yes! This can only be 9/8.

Ayşe: Where does the 9/8 come from, I couldn’t see it.

Cemile: Let me take it from scratch! We found that 6/8 is not the right answer. Because if we add

1/8 to 5/8, we’d find 6/8. Nevertheless, we have a 5/8 at hand [pointing to 5 shaded

triangles] and a 1/2 [pointing to one of the halves]. We also know that 1/2 equals to 4/8.

So if we put 4/8 and 5/8 together, the result will be 9/8.

a. What is the teacher trying to do in this scenario? Explain your answer based on the given scenario.

b. If you were the teacher in this scenario how would you handle this issue? Explain your reasoning.

****Note that the given erroneous solution method in this scenario is adopted from Kazemi and Stipek (2001).

1 2 3 4 5 6 7 8

1/2 + 1/8 + 1/8 + 1/8 + 1/8 + 1/8

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IJEMST (International Journal of Education in Mathematics, Science and Technology)

Appendix 2 – Rubrics to Characterize Participants’ Understanding of Norms

Level Norm #1 Norm #2 Norm #3 Norm #4 Norm #5

1

Very low

- Chooses one or more of the offered

solutions and explains why they are

mathematically (more) valuable.

- Does not address in any way what

perception would taking different

solutions together into consideration

develop for students. In other words,

s/he does not make any inference about

any norms.

- Discusses that all given solution methods

should be supported and encouraged.

- Chooses one or more of the given

solutions except Group #2’s solution

method and/or explains why they are

mathematically (more) valuable.

- Rejects such classroom operation using a variety of

reasons that seems to be independent of norms (e.g., not

having enough time, level of students, etc.).

- Considers students’ negotiation process in very general

terms (e.g., democratic environment, respecting ideas,

active involvement, in-class sharing, brain storming,

developing reasoning, increase of interest, etc.).

- Tie such classroom operation to be real to some

preconditions that are not closely related to norms (e.g.,

listening to each other, having a culture of discussion,

having certain math concepts as prior knowledge

classroom/student level, etc.).

Do not focus on any issue regarding

convincing or make very poor/little

explanation.

- Do not touch on that mistakes

should be taken into consideration

as a class but focuses on

mathematical aspect of the issue

and/or touches on at what point

students in the scenario make a

mistake and how s/he would

explain the answer to the students.

- Considers and values such

classroom operation with respect to

classroom participation, sharing,

brain storming etc.

2

Low

- Touches on (or not touch on) the

mathematical difference idea in the

scenario but the reason for supporting

(or not supporting) the norm is based

more on social aspects of the scenario,

and mathematical aspects are left

unattended.

- Treats the impact of different solutions

on students in very general terms [brain

storming, convenience, practical, e